Problem: Integrate. $ \int 4\sec^2(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $-4\csc(x)+C$ (Choice B) B $-4\tan(x)+C$ (Choice C) C $4\tan(x)+C$ (Choice D) D $4\csc(x)+C$
Solution: We need a function whose derivative is $4\sec^2(x)$. We know that the derivative of $\tan(x)$ is $\sec^2(x)$, so let's start there: $\dfrac{d}{dx} \tan(x) = \sec^2(x)$ Now let's multiply by $4$ : $\dfrac{d}{dx}\left[ 4\tan(x) \right] = 4\dfrac{d}{dx} \tan(x) = 4\sec^2(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 4\sec^2(x)\,dx =4 \tan(x)\, + C$ The answer: $4 \tan(x)\, + C$